In a birefringent crystal, the incident light gets split into two beams. One is ordinary and the other is extraordinary rays. The ordinary ray experiences constant refractive index n_o but the extraordinary ray experiences refractive index that varies with the direction of propagation of the light in the crystal. Why does it happen? Please explain the physical interpretation of the index ellipsoid.
Hello,
first you are using the right tool for a good understanding of propagation in anisotropic media: that is the index ellipsoid. The index ellipsoid concept is a very powerful concept as soon as you know how to use it.
1. Draw the wave vector k with its origin at the index ellipsoid center. The plane wave, perpendicular to k, will intersect the index ellipsoid as an ellipse.
2. As the index ellipsoid is linked to the direction of D vector which is perpendicular to k, a pirori any direction of D in the plane perpendicular to k is possible
3. Now consider the Maxwell's equations. They say that the vectors D, k and E are coplanar. As the plane that is tangent to the ellispoid index is orthogonal to E, you have to check which are the directions of D for which you can get k D and E in the same plane.
4. It's is quite obvious to show that only two polarizations (D vector) are allowed by the Maxwell's equations: they correspond to the big and small axes of the ellipse (see figure attached to the response - sorry it is written in French but I think that you will have not to much difficulties to understand it)
5. As seen for the extraordinary polarization, E and D are not colinear. As the Poynting vector, perpendicular to E, represents the propagation of energy, it is therefore no more colinear to k even if the vector impinges on the anisotropic medium at normal incidence. Only the incident and transmitted wave vectors are still colinear for both ordinary and extraordinary rays.
I hope it will be helpful
Look at the tensor of electrical permitivity. In the one axis crystal, in its eigenbasis two eigen values are equal i.g epsilon_11=epsilon_22. Ordinary rays are those for which the direction of electric field vector lay in the plane of two eigenvectors respective to the two, equal values of epsilon - the permitivity.
Note that for different angle of propagation, different polarizations would be considered as ordinary rays. But the value of the refractive index would be always the square root of epsilon_11=epsilon_22.
The refractive index of extraordinary rays varies since it is composed of epsilon_11 and epsilon_33 which are not equal, and their combination depends on the angle of polarization and propagation.
Hi,
I recommend reading section 8.4 In Eugene Hecht's Optics (4th edition), that explains the physical origin of birefringence based on the atomic arrangement of calcite. After reading it you'll see how a crystal is isotropic for light propagating in a specific direction, known as the optical axis. Moreover, you'll understand why calcite still posses some symmetry of rotation around this special direction, making it a uniaxial crystal and not a biaxial one. I think that after better understanding the microscopic crystal structure it will be easier for you to grasp the notion of the index ellipsoid, which simplifies the mathematical treatment of waves passing through anisotropic media but do not give any intuition regarding the physical origin of this phenomenon.
The book can be found at Amazon:
http://www.amazon.com/Optics-4th-Edition-Eugene-Hecht/dp/0805385665
I hope it helps,
Assaf
Hello,
first you are using the right tool for a good understanding of propagation in anisotropic media: that is the index ellipsoid. The index ellipsoid concept is a very powerful concept as soon as you know how to use it.
1. Draw the wave vector k with its origin at the index ellipsoid center. The plane wave, perpendicular to k, will intersect the index ellipsoid as an ellipse.
2. As the index ellipsoid is linked to the direction of D vector which is perpendicular to k, a pirori any direction of D in the plane perpendicular to k is possible
3. Now consider the Maxwell's equations. They say that the vectors D, k and E are coplanar. As the plane that is tangent to the ellispoid index is orthogonal to E, you have to check which are the directions of D for which you can get k D and E in the same plane.
4. It's is quite obvious to show that only two polarizations (D vector) are allowed by the Maxwell's equations: they correspond to the big and small axes of the ellipse (see figure attached to the response - sorry it is written in French but I think that you will have not to much difficulties to understand it)
5. As seen for the extraordinary polarization, E and D are not colinear. As the Poynting vector, perpendicular to E, represents the propagation of energy, it is therefore no more colinear to k even if the vector impinges on the anisotropic medium at normal incidence. Only the incident and transmitted wave vectors are still colinear for both ordinary and extraordinary rays.
I hope it will be helpful
Dear Sumanth,
I had already started to write down my own way of interpreting this property of the o- and e-rays, but then remembered that the corresponding wikipedia page (http://en.wikipedia.org/wiki/Birefringence) puts it in a way that is hard to beat (in its conciseness).
http://en.wikipedia.org/wiki/Birefringence
The book "Principle of Optics" (Max Born & Emil Wolf) shows the mathematical basis for this effect. The description in the book can be used for uni- and biaxial systems, if the n and the k-eikonal (n-jk: complex refractive index ) have the same orientation. If n and k are related to eikonals with different orientations, calculation is not so simple.
In an illustrative manner: Imagine a needle-shape molecule, it interacts with that component of electric field that is parallel to its axis. In an extraordinary wave this component depends on the propagation direction (projection of the T-vector on the molecule axis).
I shall say the same as Ioseph Gurwich said, but in other words:
At first necessary to understand, what is refraction index of the material and the relation between refraction index and polarization of the material. See, for example, F. L. Pedrotti and L. S. Pedrotti "Introduction to Optics", 1993, p. 568-575. There is clear physical description of interaction between light and dielectric.
If you understand the relation between refraction index and material polarizability, then it is easy to understand optical properties of different materials, particularly, of anisotropic crystals, when consider the atomic structure of the crystal (following to Radoslaw Chrapkiewicz post).
Refraction index ellipsoid is convenient, but formal tool for the description of light propagation in crystals. It does not explain physical nature of birefringence.
I hope it is correct to ask here too. I'm looking for the source giving quantatively numbers for o- and e-rays refraction indexes in the ice crystal depending on the wavelength?
https://www.researchgate.net/post/Are_broadband_refractive_indexes_of_o-and_e-rays_in_the_ice_crystal_known
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